where \(\mathbb {F}\)  is flat over \(\widetilde{\mathbf {M}}^s\) .
Let \(\tilde{\pi }_{\widetilde{\mathbf {M}}^s}:\mathcal {X}\times _{S} \widetilde{\mathbf {M}}^s\rightarrow \widetilde{\mathbf {M}}^s\)  and \(\tilde{\pi }_{\mathcal {X}}:\mathcal {X}\times _{S} \widetilde{\mathbf {M}}^s\rightarrow \mathcal {X}\)  be the projections.
Since \(\widetilde{\mathbf {M}}^s\)  is a projective \(S\) -scheme, as in [[1]}, Section 2.3] or [[2]}, Section 4.2],
we have \(R\mathcal {H}om(\mathbb {I}^{\bullet },\mathbb {I}^{\bullet })\cong \mathcal {O}_{\mathcal {X}\times _{S} \widetilde{\mathbf {M}}^s}\oplus R\mathcal {H}om(\mathbb {I}^{\bullet },\mathbb {I}^{\bullet })_{0}\) , where \(R\mathcal {H}om(\mathbb {I}^{\bullet },\mathbb {I}^{\bullet })_{0}\)  denotes its traceless part (see [[3]}, Section 3] for the trace map).
Let \(\mathbb {L}_{\mathcal {X}\times _{S} \widetilde{\mathbf {M}}^s/S}\)  be the truncated cotangent complex, i.e., the truncation \(\tau ^{\ge -1}L^\bullet _{\mathcal {X}\times _{S} \widetilde{\mathbf {M}}^s/S}\)  of Illusie's cotangent complex \(L^\bullet _{\mathcal {X}\times _{S} \widetilde{\mathbf {M}}^s/S}\) . Then we have the truncated Atiyah class of \(\mathbb {I}^{\bullet }\)
\(\mathrm {At}(\mathbb {I}^{\bullet })\in \mathrm {Ext}^1(\mathbb {I}^{\bullet },\mathbb {I}^{\bullet }\otimes \mathbb {L}_{\mathcal {X}\times _{S} \widetilde{\mathbf {M}}^s/S})\) 
